The law of sines is a trigonometric rule that links the sides of any triangle to the sines of their opposite angles: asin⁡A=bsin⁡B=csin⁡C\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}sinAa​=sinBb​=sinCc​. It is mainly used to solve non‑right (oblique) triangles when certain combinations of sides and angles are known.

What the law of sines says

  • In any triangle with sides a,b,ca,b,ca,b,c and opposite angles A,B,CA,B,CA,B,C, the ratio “side ÷ sine of opposite angle” is the same for all three sides: asin⁡A=bsin⁡B=csin⁡C\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}sinAa​=sinBb​=sinCc​.
  • This common ratio is equal to 2R2R2R, where RRR is the radius of the triangle’s circumcircle (the unique circle passing through all three vertices).

When to use the law of sines

  • It works best when you know:
    • Two angles and one side (AAS or ASA), or
    • Two sides and a non‑included angle (SSA), which can sometimes lead to the “ambiguous case” with two possible triangles.
  • It is not the main tool when all three sides or two sides and the included angle are known; in those cases, the law of cosines is usually more appropriate.

A quick example

  • Suppose a=7a=7a=7, A=60∘A=60^\circ A=60∘, and B=45∘B=45^\circ B=45∘; using the law of sines,
    asin⁡A=bsin⁡B\dfrac{a}{\sin A}=\dfrac{b}{\sin B}sinAa​=sinBb​ ⇒ 7sin⁡60∘=bsin⁡45∘\dfrac{7}{\sin 60^\circ}=\dfrac{b}{\sin 45^\circ}sin60∘7​=sin45∘b​, which lets you solve for the unknown side bbb.
  • This illustrates how the law is used to “solve a triangle” by finding missing sides and angles once enough information is known.

HTML table: key facts

html

<table>
  <thead>
    <tr>
      <th>Aspect</th>
      <th>Law of sines detail</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>Core formula</td>
      <td>a/sin A = b/sin B = c/sin C = 2R[web:1][web:3]</td>
    </tr>
    <tr>
      <td>Triangle type</td>
      <td>Applies to any (oblique or right) triangle[web:1][web:5][web:7]</td>
    </tr>
    <tr>
      <td>Best use cases</td>
      <td>AAS, ASA, or SSA configurations to find unknown sides/angles[web:3][web:6][web:7]</td>
    </tr>
    <tr>
      <td>Ambiguous case</td>
      <td>SSA can produce 0, 1, or 2 possible triangles[web:4][web:6]</td>
    </tr>
    <tr>
      <td>Connection to circle</td>
      <td>Common ratio equals diameter of circumcircle (2R)[web:1][web:3]</td>
    </tr>
  </tbody>
</table>

Information gathered from public forums or data available on the internet and portrayed here.